840 lines
64 KiB
Plaintext
840 lines
64 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "4cc6c96f",
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"metadata": {},
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"source": [
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"# Ćwiczenia 6"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1276423e",
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"metadata": {},
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"source": [
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"***TEMAT:*** funkcje wielu zmiennych - definicja, wykres, granica, ciągłość, pochdone cząstkowe i gradient"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a92c3fcc",
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"metadata": {},
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"source": [
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"## Funkcje wielu zmiennych: definicja i wykresy niektórych funkcji"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c85e4fed-46e6-4310-9128-337d82b8c235",
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"metadata": {},
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"source": [
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"Funkcją $n$-zmiennych ($n \\geq 2$) określoną na zbiorze $A\\subset\\mathbb{R}^n$ o wartościach w $\\mathbb{R}$ nazywamy przyporządkowanie każdemu punktowi ze zbioru $A$ dokładnie jednej liczby rzeczywistej. Funkcję taką oznaczamy przez $f:A\\rightarrow \\mathbb{R}$. Wartość funkcji $f$ w punkcie $(x_1, ... x_n)$ oznaczamy przez $f(x_1, ... x_n)$."
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]
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},
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{
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"cell_type": "markdown",
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"id": "b0897f5c-381a-47df-8e6a-e20b5b665b05",
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"metadata": {},
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"source": [
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"#### Zadanie 1 \n",
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"\n",
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"Wyznacz dziedzinę następujących funkcji i naszkicuj ją na płaszczyźnie:\n",
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">1. $f(x,y)=\\sqrt{4-(x-4)^2-(y+2)^2}$;\n",
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">2. $f(x,y)=\\ln x\\cdot \\ln y +\\frac{1}{\\sqrt{1-x-y}}$."
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]
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},
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{
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"cell_type": "markdown",
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"id": "303dc23a-6e38-4435-b067-bfe1d99742f5",
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"metadata": {},
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"source": [
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"##### Rozwiązanie:"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7ae3148c",
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"metadata": {},
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"source": [
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"1. Dziedzinę funkcji $f$ tworzy zbiór tych $(x,y)$ takich, że \n",
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"\n",
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"$$\n",
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"4-(x-4)^2-(y+2)^2\\geq 0\n",
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"$$\n",
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"\n",
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"czyli zbiór rozwiązań nierówności\n",
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"\n",
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"$$\n",
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"(x-4)^2+(y+2)^2\\leq 4.\n",
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"$$\n",
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"\n",
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"Jest to kolo o środku w punkcie $(4,-2)$ i promieniu $2$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "d9fd7f28",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": "iVBORw0KGgoAAAANSUhEUgAAAcAAAAHKCAYAAACUvpsVAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjguNCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8fJSN1AAAACXBIWXMAAA9hAAAPYQGoP6dpAAAocElEQVR4nO3df3RU5Z3H8U8Iw4QAQZMcIIFAAlIDpQgEf4D8CCooUhdxly0WFZGyiEChnFWodEuwQBABPSsVBS3aogvbohVFXFJNQOUgMUSW2jUoEEASiqImQHAYkmf/sMkh5AcJzOTOnef9OmeOzjN37ny/zx3uJ8/MJBNhjDECAMAyzZwuAAAAJxCAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYBAgPh8Pk2cOFFJSUmKiYnRDTfcoB07djhdluswj2gqBCAQIOfOnVNKSoo++OADffvtt5o6dar+6Z/+SWVlZU6X5irMI5pKBN8GAQRPbGyssrOzdc011zhdiqsxjwgGVoBw1GOPPaaePXuqoqLC6VKqef755xUREaHWrVvXuO2FF15Qx44ddfr06Xr38emnn+rMmTPq1q1bsMqsEkrz+O677+qBBx5QamqqWrVqpY4dO2r06NHKy8urtl0ozmOl+o4/wogBHHL06FHTqlUr88c//tHpUqr54osvTNu2bU1iYqJp1apVjdv9fr/p3r27+fWvf13nPk6fPm369+9vFi5cGMxSjTGhN4//8i//YoYNG2aeeeYZk5OTY/74xz+aG264wTRv3ty88847VduF2jxWutjxR/jgJVA4Zs6cOXr55Zd1+PBhNWsWOi9G3HHHHYqIiFBsbKz+9Kc/6dSpUzW2Wb58uX7zm9+oqKhI0dHR1W7z+/0aM2aM4uLi9OKLLyoiIiKo9QZ7HtPT05WcnKwXX3yxQdsfP35c7dq1qzZ26tQpXXXVVerVq5f+8pe/VI03xTxWVFRo165d2rJliyZOnKjk5OR6t2/I8Ud4CJ2zDqxy9uxZvfDCC/rpT39a7aRdXFys1q1ba9y4cdW2f/PNN+XxeDRv3ryg1rVu3Tpt27ZNzzzzTL3bjR8/XqWlpVq/fn218YqKCt13332KjIzUCy+8EPTwq2se77rrLnXq1KnG9ufOnVOfPn00fPjwoNV0YfhJUuvWrdWzZ08dOXKk2niw5vHLL7/UunXrNH78eLVr104DBgzQb3/7W507d67e+zX0+CM8EIBwxIcffqgTJ05o2LBh1cYTEhL0yCOP6L//+7+r3jPKycnR2LFjNXXqVC1atChoNR0/flyzZs3SkiVLag2P83Xo0EGpqanavHlztfEpU6aouLhYGzZsUPPmzYNWa6W65nHIkCE6evSoDh06VG18xYoV+vTTT5v8BF9SUqLdu3frhz/8YbXxQM1jRUWFPvzwQ2VkZOi6665T+/btdd9996mgoEBTp07Vjh07dPz4cV111VV17qMxxx9hwunXYGGnxx9/3Egyx44dq3Hb6dOnTWJiorn55pvNrl27TJs2bczEiRNNRUVFUGv653/+ZzNw4MCqx5kwYUK97wGNHz/etG/fvup6YWGhkWSioqJMq1atqi7bt28PWs11zWNeXp6RZF555ZWqsQMHDpjo6Gjz2GOP1bm/iooK4/f7q12GDBli7rvvvhrjjTF+/HjTvHlz89FHH9V62+XMY35+vomLizOSzBVXXGH+9V//1axdu7bW51Z9Gnv84X6sAOGIoqIiRUREKD4+vsZt0dHRWrhwod555x0NGzZMI0eO1Jo1a+p8GSwnJ0cRERENunz88ce17mPjxo1644036n2cC7Vr107Hjx+velmtS5cuMsbozJkzOnXqVNVl8ODBQau7rnns06ePYmJi9MEHH1SNTZ06VUlJSZozZ06dPW3btk0ej6faZfv27fr9739fY7ywsLBB8/Qf//Efevnll/Xkk08qLS2txu2XO4+RkZFV7x+eO3dO3333XdWloS7l+MP9gv8aDVCLM2fOyOPxKDIystbbf/CDH0iSIiIi9OKLL9a5nSRdffXVWrNmTYMet3PnzjXGTp06pWnTpmnGjBlKTEzUt99+K+n799ck6dtvv5XH41GrVq2q3S8qKkrGGH333XeX9HH5y61bqnsemzVrpoEDB1b9BZWXX35Z//M//6Ps7Gy1aNGizsdJS0tTbm5utbEpU6YoMTFR8+fPrzaemJh40boXLFighQsXatGiRZo+fXqt21zuPP7oRz/S4cOH9de//lVbtmzRW2+9pZ///Ofy+/3q2bOnRo4cqZEjR2rw4MG19n6pxx9hwNkFKGz16KOPGknm1KlTNW7Lz883V1xxhbnxxhuNJLNy5cqg1nLw4EEjqd7L6NGja9xvypQpxuv1BrW2i6lvHhcvXmwiIyPNoUOHTLt27cyECRMu6TGGDh16SffNyMgwkkxGRka92wVjHktLS82rr75qJk+ebDp16mQkmdatW5uCgoIa217q8Yf7sQKEI1JTUyVJ+/fvV+/evavGCwoKdOutt2rAgAF6/fXXNXbsWGVkZOiee+5R27Ztg1JLhw4dlJ2dXWN8yZIl2rZtm7Zs2VLrS7UHDhxQz549g1JTQ9U1j9L3H4QpLy/Xj3/8Y5WXl2vZsmVNVtdvfvMbZWRk6Fe/+lWNleOFgjGPbdq00ZgxYzRmzBhJ0t69e7VlyxZFRUXV2PZSjz/cjwCEI9LT0yVJO3furDpxFxYW6pZbbtHVV1+tjRs3yuPxaMmSJerVq5cWL16sxx9/PCi1REVFVdVzvsqXXmu7rfJ3yyZNmhSUmhqqtnmsdO2116ply5bau3evfve73zXZSXz58uX69a9/rdtuu02jRo3Szp07q91+ww03VP1/IOaxuLi4xqdIaxMbG1vrS6yXcvwRJpxegsJegwcPNrfffrsxxpiioiLTrVs3069fP1NSUlJtu8mTJxuv12sOHjzYpPXV9ynAd955x0gyeXl5TVpTbc6fx/P5/X6TmJhoBg8efFmfoG3sS6BDhw6t9+XE8wViHrOzsy/6EmblJT8/v8H75VOg4Y+/BAPHbNy4UT/5yU906NAhdezY0elyGuXee+/VgQMHqn3K0il1zeOyZcs0b948ffzxx+rRo4eDFdYtlOYR9iEA4RhjjAYOHKi0tDStXLnS6XIabP/+/erRo4feffddDRo0yOlyqs3j0qVLtWfPHuXm5urhhx/WokWL9O///u9Ol1irUJtH2IffA4RjIiIitGbNGiUmJobEtxg01OHDh7Vy5cqQOWmfP49vv/22Bg4cqMzMTD366KMhG35S6M0j7MMKEABgJdeuADMzMxUREaFZs2Y5XQoAwIVcGYC5ublavXp1jY99AwDQUK4LwFOnTmn8+PFas2aNrrzySqfLAQC4lOt+EX7atGkaNWqUbrnlFi1cuLDebX0+n3w+X9X1iooKff3114qLi+MP3gKAyxhjdPLkSSUmJgbky59dFYDr16/X7t27a/yx3rpkZmZqwYIFQa4KANCUjhw5EpDvbHRNAB45ckQzZ87U1q1ba/17frX55S9/qdmzZ1ddLykpUefOnbVv3z7FxsYGq9SQ4vf7NWrUKG3evFkej8fpcpqM3+9Xdna2hg0bRt9hzsaeJTv7/vrrr/WDH/xAbdq0Ccj+XBOAeXl5On78eLXvEysvL9f27du1cuVK+Xy+Gl8J4/V65fV6a+wrNjZWcXFxQa85FPj9fkVGRiouLs6afyTS931HR0fTtwVs7Fmyt29JAXsLyzUBePPNN2vv3r3VxiZOnKjU1FTNmTOn3u+LAwDgQq4JwDZt2qhXr17Vxlq1aqW4uLga4wAAXIzrfg0CAIBAcM0KsDY5OTlOlwAAcClWgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACsRgAAAKxGAAAArEYAAACs
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"text/plain": [
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"<Figure size 640x480 with 1 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"\n",
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"# Define the grid of x and y values\n",
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"x = np.linspace(-2, 10, 400)\n",
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"y = np.linspace(-8, 4, 400)\n",
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"X, Y = np.meshgrid(x, y)\n",
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"\n",
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"# Define the function for the inequality (x-4)^2 + (y+2)^2 <= 4\n",
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"Z = (X - 4)**2 + (Y + 2)**2 - 4\n",
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"\n",
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"# Plot the contour for the region where Z <= 0\n",
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"plt.contourf(X, Y, Z, levels=[-1000, 0], colors=['lightblue'])\n",
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"\n",
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"# Additional plot settings\n",
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"plt.xlabel('x')\n",
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"plt.ylabel('y')\n",
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"plt.title(r'$(x-4)^2 - (y+2)^2 \\leq 4$')\n",
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"plt.axhline(0, color='black',linewidth=0.5)\n",
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"plt.axvline(0, color='black',linewidth=0.5)\n",
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"plt.grid(True)\n",
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"plt.gca().set_aspect('equal', adjustable='box')\n",
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"plt.show()\n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "c4050694",
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"metadata": {},
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"source": [
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"2. Dziedzinę funkcji $f$ tworzy zbiór tych $(x,y)$ dla których\n",
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"\n",
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"$$\n",
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"x>0, y>0 \\text{ oraz } 1-x-y>0.\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "3b933612",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": "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
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"import numpy as np\n",
|
||
|
|
"import matplotlib.pyplot as plt\n",
|
||
|
|
"\n",
|
||
|
|
"# Define the x range for the plot\n",
|
||
|
|
"x = np.linspace(0, 1, 400)\n",
|
||
|
|
"\n",
|
||
|
|
"# Define the boundary lines\n",
|
||
|
|
"y1 = 0 * x # y = 0 line for y > 0\n",
|
||
|
|
"y2 = 1 - x # y = 1 - x line for 1 - x - y > 0\n",
|
||
|
|
"\n",
|
||
|
|
"# Fill the region where the inequalities hold\n",
|
||
|
|
"plt.fill_between(x, y1, y2, where=(y2 > y1), color=\"lightblue\", alpha=0.5)\n",
|
||
|
|
"\n",
|
||
|
|
"# Plot the boundaries with dashed lines\n",
|
||
|
|
"plt.plot(x, y2, 'k--', label=r'$1 - x - y = 0$')\n",
|
||
|
|
"plt.axvline(0, color='k', linestyle='--', label=r'$x = 0$')\n",
|
||
|
|
"plt.axhline(0, color='k', linestyle='--', label=r'$y = 0$')\n",
|
||
|
|
"\n",
|
||
|
|
"# Additional plot settings\n",
|
||
|
|
"plt.xlim(-0.1, 1.1)\n",
|
||
|
|
"plt.ylim(-0.1, 1.1)\n",
|
||
|
|
"plt.xlabel('x')\n",
|
||
|
|
"plt.ylabel('y')\n",
|
||
|
|
"plt.title(r'Set: $x > 0$, $y > 0$, $1 - x - y > 0$')\n",
|
||
|
|
"plt.grid(True)\n",
|
||
|
|
"plt.legend(loc=\"upper right\")\n",
|
||
|
|
"plt.gca().set_aspect('equal', adjustable='box')\n",
|
||
|
|
"plt.show()\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "7b9026af-9d58-490c-8707-154352d22ce3",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 2 \n",
|
||
|
|
"Wykreśl wykres poniższych funkcji:\n",
|
||
|
|
" >1. $f(x,y)=x^2+y^2$;\n",
|
||
|
|
" >2. $f(x,y)=\\sqrt{x^2+y^2}$;\n",
|
||
|
|
" >3. $f(x,y)=\\sqrt{16-x^2-y^2}$ i $g(x,y)=-\\sqrt{16-x^2-y^2} $;\n",
|
||
|
|
" >4. $f(x,y)=x^2-y^2$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "fdfcf3f2-447a-42af-ad58-82ebe18b8218",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"##### Rozwiązanie:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 12,
|
||
|
|
"id": "1a1d6417-4095-4f3b-a020-cab300fd9de6",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"import IPython.display as display\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 1,
|
||
|
|
"id": "5ccf805e-2bc0-4b70-a62a-a92254ea9c70",
|
||
|
|
"metadata": {
|
||
|
|
"scrolled": true
|
||
|
|
},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"\n",
|
||
|
|
" <iframe\n",
|
||
|
|
" width=\"800\"\n",
|
||
|
|
" height=\"600\"\n",
|
||
|
|
" src=\"https://www.geogebra.org/calculator/ahx8sscp?style=border%3A+1px+solid+black\"\n",
|
||
|
|
" frameborder=\"0\"\n",
|
||
|
|
" allowfullscreen\n",
|
||
|
|
" \n",
|
||
|
|
" ></iframe>\n",
|
||
|
|
" "
|
||
|
|
],
|
||
|
|
"text/plain": [
|
||
|
|
"<IPython.lib.display.IFrame at 0x7f1a946c9af0>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 1,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"from IPython.display import IFrame\n",
|
||
|
|
"IFrame('https://www.geogebra.org/calculator/ahx8sscp', width=800, height=600, style=\"border: 1px solid black\")"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 2,
|
||
|
|
"id": "08cee401-113a-477e-bb0a-1bfa51afd412",
|
||
|
|
"metadata": {
|
||
|
|
"scrolled": true
|
||
|
|
},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"\n",
|
||
|
|
" <iframe\n",
|
||
|
|
" width=\"800\"\n",
|
||
|
|
" height=\"600\"\n",
|
||
|
|
" src=\"https://geogebra.org/calculator/x4ngbw4r?style=border%3A+1px+solid+black\"\n",
|
||
|
|
" frameborder=\"0\"\n",
|
||
|
|
" allowfullscreen\n",
|
||
|
|
" \n",
|
||
|
|
" ></iframe>\n",
|
||
|
|
" "
|
||
|
|
],
|
||
|
|
"text/plain": [
|
||
|
|
"<IPython.lib.display.IFrame at 0x7f79d050d850>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 2,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"from IPython.display import IFrame\n",
|
||
|
|
"IFrame('https://geogebra.org/calculator/x4ngbw4r', width=800, height=600, style=\"border: 1px solid black\")"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 3,
|
||
|
|
"id": "2f7a5f9f-b234-416f-af00-9f92eb55c53d",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"\n",
|
||
|
|
" <iframe\n",
|
||
|
|
" width=\"800\"\n",
|
||
|
|
" height=\"600\"\n",
|
||
|
|
" src=\"https://www.geogebra.org/calculator/dywfgwgb?style=border%3A+1px+solid+black\"\n",
|
||
|
|
" frameborder=\"0\"\n",
|
||
|
|
" allowfullscreen\n",
|
||
|
|
" \n",
|
||
|
|
" ></iframe>\n",
|
||
|
|
" "
|
||
|
|
],
|
||
|
|
"text/plain": [
|
||
|
|
"<IPython.lib.display.IFrame at 0x7f79d050d9a0>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 3,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"from IPython.display import IFrame\n",
|
||
|
|
"IFrame('https://www.geogebra.org/calculator/dywfgwgb', width=800, height=600, style=\"border: 1px solid black\")"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 4,
|
||
|
|
"id": "896bc403-9945-4d66-a905-9315debb3abd",
|
||
|
|
"metadata": {
|
||
|
|
"scrolled": true
|
||
|
|
},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
"text/html": [
|
||
|
|
"\n",
|
||
|
|
" <iframe\n",
|
||
|
|
" width=\"800\"\n",
|
||
|
|
" height=\"600\"\n",
|
||
|
|
" src=\"https://www.geogebra.org/calculator/rztxuac3?style=border%3A+1px+solid+black\"\n",
|
||
|
|
" frameborder=\"0\"\n",
|
||
|
|
" allowfullscreen\n",
|
||
|
|
" \n",
|
||
|
|
" ></iframe>\n",
|
||
|
|
" "
|
||
|
|
],
|
||
|
|
"text/plain": [
|
||
|
|
"<IPython.lib.display.IFrame at 0x7f79d04ff160>"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 4,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"from IPython.display import IFrame\n",
|
||
|
|
"IFrame('https://www.geogebra.org/calculator/rztxuac3', width=800, height=600, style=\"border: 1px solid black\")"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "05e2b5c1-e621-4c3e-bea3-15c142af68c4",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"## Granica i ciągłość"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "13dd02b5",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"### Granica\n",
|
||
|
|
"Niech $A\\subset \\mathbb{R}^n$ i niech $x_0\\in A$ będzie punktem skupienia zbioru $A$ (to oznacza, że $x_0$ jest granicą ciągu $(x_n)$ punktów ze zbioru $A\\setminus\\{x_0\\}$). Mówimy, że funkcja $f:A\\to\\mathbb{R}$ ma granicę w punkcie $x_0$ równą $g$ i piszemy\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\lim_{x\\to x_0 } f(x)=g,\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"gdy dla każdego ciągu $(x_n)$ punktów zbioru $A\\setminus\\{x_0\\}$ zachodzi\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"x_n \\xrightarrow{n \\to \\infty} x_0\\Rightarrow f(x_n) \\xrightarrow{n \\to \\infty} g.\n",
|
||
|
|
"$$"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "a7630640-71b9-46be-a910-32d41d26d7d6",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 3 \n",
|
||
|
|
"\n",
|
||
|
|
"Oblicz następujące granice\n",
|
||
|
|
">1. $\\displaystyle \\lim_{(x,y)\\to (1,1)}\\frac{x^2y}{x^2+y^2}$;\n",
|
||
|
|
">2. $\\displaystyle \\lim_{(x,y)\\to (0,0)}\\frac{x^2y}{x^2+y^2}$;\n",
|
||
|
|
">3. $\\displaystyle \\lim_{(x,y)\\to (0,0)}\\frac{x^2+y^2}{\\sqrt{x^2+y^2}}$;\n",
|
||
|
|
">4. $\\displaystyle \\lim_{(x,y)\\to (0,0)}\\frac{x^2y^3}{x^4+y^4}$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "4b48884a-2d2d-4994-925d-2281759c470f",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"##### Rozwiązanie:\n",
|
||
|
|
"1. Z własności granic bardzo łatwo dostajemy, że \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\displaystyle \\lim_{(x,y)\\to (1,1)}\\frac{x^2y}{x^2+y^2}=\\frac{1}{2}.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"2. Dla $(x,y)\\not=0$ mamy \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\left|\\frac{x^2y}{x^2+y^2}\\right|\\leq |y|\\cdot \\frac{x^2}{x^2+y^2}\\leq |y|.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Zatem \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\lim_{(x,y)\\to (0,0)}\\frac{x^2y}{x^2+y^2}=0.\n",
|
||
|
|
"$$\n",
|
||
|
|
"3. Dla $(x,y)\\not=(0,0)$ mamy\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\left|\\frac{x^2+y^2}{\\sqrt{x^2+y^2}}\\right|\\leq |x|\\cdot\\frac{|x|}{\\sqrt{x^2+y^2}}+|y|\\cdot\\frac{|y|}{\\sqrt{x^2+y^2}}\\leq |x|+|y|.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Zatem\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\lim_{(x,y)\\to (0,0)}\\frac{x^2+y^2}{\\sqrt{x^2+y^2}}=0.\n",
|
||
|
|
"$$\n",
|
||
|
|
"4. Niech dla $(x,y)\\not=(0,0)$ \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(x,y)=\\frac{x^2y^3}{x^4+y^4},\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"niech\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"(0,0)\\not=(x_n,y_n)\\xrightarrow{n\\to\\infty}(0,0)\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"i niech \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"u_n=\\max\\{|x_n|,|y_n|\\}.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Jest jasne, że \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"u_n\\xrightarrow{n\\to\\infty}0.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Ponadto \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"|f(x_n,y_n)|=\\left|\\frac{x_n^2y_n^3}{x_n^4+y_n^4}\\right|\\leq\\frac{u_n^5}{u_n^4}=u_n.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Zatem \n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(x_n,y_n)\\xrightarrow{n\\to\\infty}0.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Ostatecznie \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\lim_{(x,y)\\to (0,0)}\\frac{x^2y^3}{x^4+y^4}=0.\n",
|
||
|
|
"$$"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "d10f6a21-3419-4d0e-9bc6-67286fab908f",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 4\n",
|
||
|
|
"Pokaż, że nastepujace funkcje nie mają granicy w punkcie (0,0).\n",
|
||
|
|
"\n",
|
||
|
|
">1. $f(x,y)=1$ jeśli $x=0$ lub $y=0$ oraz $f(x,y)=0$ w przeciwnym wypadku.\n",
|
||
|
|
">2. $\\displaystyle f(x,y)=\\frac{xy}{x^2+y^2}$ dla $(x,y)\\not=0$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "fee6a963",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"##### Rozwiązanie:\n",
|
||
|
|
"\n",
|
||
|
|
"1. Ciągi $(0,1/n)_{n\\in\\mathbb{N}}$ oraz $(1/n,1/n)_{n\\in\\mathbb{N}}$ dążą do punktu $(0,0)$. \n",
|
||
|
|
"Ponadto \n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(0,1/n)=1\\xrightarrow{n\\to\\infty}1 \\quad\\text{ i }\\quad f(1/n,1/n)=0\\xrightarrow{n\\to\\infty}0. \n",
|
||
|
|
"$$\n",
|
||
|
|
"Zatem funkcja $f$ nie ma granicy w punkcie $(0,0)$.\n",
|
||
|
|
"\n",
|
||
|
|
"2. Ciągi $(0,1/n)_{n\\in\\mathbb{N}}$ oraz $(1/n,1/n)_{n\\in\\mathbb{N}}$ dążą do punktu $(0,0)$. \n",
|
||
|
|
"Ponadto \n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(0,1/n)=0\\xrightarrow{n\\to\\infty}0\\quad \\text{ i }\\quad f(1/n,1/n)=1/2\\xrightarrow{n\\to\\infty}1/2. \n",
|
||
|
|
"$$\n",
|
||
|
|
"Zatem funkcja $f$ nie ma granicy w punkcie $(0,0)$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "2076a8a2",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"### Ciągłość\n",
|
||
|
|
"\n",
|
||
|
|
"Niech $A\\subset \\mathbb{R}^n$ i niech $f:A\\to\\mathbb{R}$. Mówimy, że funkcja $f$ jest ciągła w punkcie $x_0\\in A$, gdy \n",
|
||
|
|
" dla każdego ciągu $(x_n)$ punktów zbioru $A$ zachodzi\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"x_n \\xrightarrow{n \\to \\infty} x_0\\Rightarrow f(x_n) \\xrightarrow{n \\to \\infty} f(x_0).\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"***Uwaga:***\n",
|
||
|
|
">* Jeśli $x_0$ **nie** jest punktem skupienia zbioru $A$, to funkcja $f$ **jest** ciągła w punkcie $x_0$.\n",
|
||
|
|
">* Jeśli $x_0$ jest punktem skupienia zbioru $A$, to funkcja $f$ **jest** ciągła w punkcie $x_0$ dokładnie wtedy gdy \n",
|
||
|
|
"$\\displaystyle \\lim_{x\\to x_0}f(x)=f(x_0)$.\n",
|
||
|
|
">* Jeśli funkcja $f$ jest ciągła w każdym punkcie zbioru $A$, to mówimy że jest ciągła na $A$.\n",
|
||
|
|
"\n",
|
||
|
|
"\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "ab4a522c-652e-461d-a880-af51d80b94f8",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 5\n",
|
||
|
|
"\n",
|
||
|
|
"Omów ciągłość poniższych funkcji.\n",
|
||
|
|
"\n",
|
||
|
|
">1. $f(x,y,z)=x^2y+z$.\n",
|
||
|
|
">2. $f(x,y)=\\frac{x^4}{x^2+y^2}$, gdy $(x,y)\\not=(0,0)$ oraz $f(0,0)=1$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "ce59d7b8",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"##### Rozwiązanie:\n",
|
||
|
|
"1. Jest jasne, że funkcja $f$ jest funkcją ciągłą na $\\mathbb{R}^3$. Istotnie, niech \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"(x_n,y_n,z_n)\\xrightarrow{n\\to\\infty}(x_0,y_0,z_0).\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Wtedy \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"x_n\\xrightarrow{n\\to\\infty}x_0,\\quad y_n\\xrightarrow{n\\to\\infty}y_0,\\quad z_n\\xrightarrow{n\\to\\infty}z_0.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Zatem \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(x_n,y_n,z_n)=x_n^2y_n+z_n\\xrightarrow{n\\to\\infty}x^2_0y_0+z_0=f(x_0,y_0,z_0).\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"2. Argumenty takie jak wyżej i wiedza z wykłady łatwo dają, że funkcja $f$ jest ciągła na zbiorze $\\mathbb{R}^2\\setminus\\{(0,0)\\}$.\n",
|
||
|
|
"Mamy\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(0,1/n)=0.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Zatem $f$ nie jest ciągła w punkcie $(0,0)$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "933b8b8b-4b65-4cf0-99ac-a07e67c0041e",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"## Pochodne cząstkowe"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "b8047623-9367-4eca-b8a1-bae341bdcc59",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"### Definicja i podstawowe rachunki"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "a4098dbc-47a6-475d-aed3-4d9a130c31f2",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Pochodną cząstkową I rzędu funkcji $f$ w punkcie $(x_1, ... , x_n)$ względem zmiennej $x_k$ nazywamy granicę ilorazu różnicowego:\n",
|
||
|
|
"\n",
|
||
|
|
"$$ \n",
|
||
|
|
"{\\frac{\\partial f}{\\partial x_k}(x_1, ..., x_n):=\\lim\\limits_{h \\to 0 } \\frac{f(x_1, ... , x_{k-1} , x_k+h, x_{k+1}, ... x_n) - f(x_1, ..., x_n)}{h}.}\n",
|
||
|
|
"$$\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "d446b8ec-7b97-4a08-b7af-d04c59d9a610",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 5 \n",
|
||
|
|
"\n",
|
||
|
|
"Oblicz wszystkie pochodne cząstkowe I rzędu dla funkcji \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(x,y,z)=xy^2+z.\n",
|
||
|
|
"$$"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "bd6fe862",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"##### Rozwiązanie:\n",
|
||
|
|
"1.\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial x}(x,y,z)= \\lim_{h\\to 0} \\frac{f((x+h,y,z))-f(x,y,z)}{h}=\n",
|
||
|
|
"\\lim_{h\\to 0}\\frac{hy^2}{h}=y^2.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"2.\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial y}(x,y,z)= \\lim_{h\\to 0} \\frac{f((x,y+y,z))-f(x,y,z)}{h}=\n",
|
||
|
|
"\\lim_{h\\to 0}\\frac{x(y+h)^2-xy^2}{h}=\n",
|
||
|
|
"=\\lim_{h\\to 0}\\frac{2xyh+xh^2}{h}=\\lim_{h\\to 0}(2xy+xh)=2xy.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"3.\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial z}(x,y,z)= \\lim_{h\\to 0} \\frac{f((x,y,z+h))-f(x,y,z)}{h}=\n",
|
||
|
|
"\\lim_{h\\to 0}\\frac{h}{h}=1.\n",
|
||
|
|
"$$\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "dd16f502",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 6\n",
|
||
|
|
"\n",
|
||
|
|
"Obliczymy pochodne cząstkowe funkcji \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(x,y)=\\begin{cases}\n",
|
||
|
|
"\\frac{xy}{x^2+y^2}, & (x,y)\\not=(0,0)\\\\\n",
|
||
|
|
"0, & (x,y)=(0,0).\n",
|
||
|
|
"\\end{cases}\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"w punkcie $(0,0)$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "3b0b301c",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"##### Rozwiązanie:"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "4c2c9446",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Mamy\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial x}(0,0)=\\lim_{h\\to 0}\\frac{f(h,0)-f(0,0)}{h}\n",
|
||
|
|
"=\\lim_{h\\to 0}\\frac{0-0}{h}=0\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"oraz\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial y}(0,0)=\\lim_{h\\to 0}\\frac{f(0,h)-f(0,0)}{h}\n",
|
||
|
|
"=\\lim_{h\\to 0}\\frac{0-0}{h}=0.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"***Uwaga:*** zauważmy, że \n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"f(1/n,1/n)=1/2\\xrightarrow{n\\to\\infty}1/2\\not=0=f(0,0).\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Stąd $f$ nie jest ciągła w punkcie $(0,0)$. Zatem istnienie pochodnych cząstkowych w punkcie $(0,0)$ __nie implikuje__ ciągłości funkcji w tym punkcie"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "44450f43-6d68-44ca-ac06-909841547ba1",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 7\n",
|
||
|
|
"\n",
|
||
|
|
"Zbadaj istnienie pochodnych cząstkowych funkcji $f(x,y)=\\sqrt{x^2+y^4}$ w punkcie $(0,0)$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "e0da843f",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"##### Rozwiązanie:\n",
|
||
|
|
"\n",
|
||
|
|
"1. Mamy\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial x}(0,0)=\\lim_{h\\to 0}\\frac{f(h,0)-f(0,0)}{h}\n",
|
||
|
|
"=\\lim_{h\\to 0}\\frac{|h|}{h}.\n",
|
||
|
|
"$$\n",
|
||
|
|
"\n",
|
||
|
|
"Ta granica nie istnieje, zatem funkcja $f$ nie ma pochodnej cząstkowej względem pierwszej zmiennej w punkcie $(0,0)$.\n",
|
||
|
|
"\n",
|
||
|
|
"2. Na podstawie definicji otrzymujemy, że \n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial y}(0,0)=\\lim_{h\\to 0}\\frac{f(0,h)-f(0,0)}{h}\n",
|
||
|
|
"=\\lim_{h\\to 0}\\frac{h^2}{h}=0.\n",
|
||
|
|
"$$\n"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "36fc2bcf-a2dd-47d1-abbb-06bd3782e2bb",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"#### Zadanie 8 "
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "67fd2093",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Korzystając z reguł różniczkowania oblicz pochodne cząstkowe poniższych funkcji.\n",
|
||
|
|
"\n",
|
||
|
|
"> 1. $f(x,y,z)=x^2+2xyz+yz^3$;\n",
|
||
|
|
"> 2. $f(x,y)=xe^{x^2+y^2}$;\n",
|
||
|
|
"> 3. $f(x,y)=\\frac{x^2+y}{x^2+y^2+1}$."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "346feb36-5e77-4e51-9435-df2ea46f3517",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Dla $f(x,y,z) = x^2 + 2xyz + yz^3$:\n",
|
||
|
|
"\n",
|
||
|
|
"$$\n",
|
||
|
|
" \\frac{\\partial f}{\\partial x}{(x, y, z)} = 2x + 2yz\n",
|
||
|
|
"$$\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial y}{(x, y, z)} = 2xz + z^3\n",
|
||
|
|
"$$\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial z} {(x, y, z)} = 2xy + 3yz^2\n",
|
||
|
|
"$$"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "0027c7c4-460f-42fa-91ce-d04330e3152d",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Dla $f(x,y) = x e^{x^2 + y^2}$:\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial x}{(x, y)} = e^{x^2 + y^2} + 2x^2 e^{x^2 + y^2}\n",
|
||
|
|
"$$\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial y}{(x, y)} = 2x y e^{x^2 + y^2}\n",
|
||
|
|
"$$"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"id": "81d36af3-1408-4345-8ec1-7f107b5189c1",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Dla $ f(x,y) = \\frac{x^2 + y}{x^2 + y^2 + 1} $:\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial x} {(x, y)} = \\frac{2x(x^2 + y^2 + 1) - (x^2 + y)(2x)}{(x^2 + y^2 + 1)^2} = \\frac{2x(x^2 + y^2 + 1 - y)}{(x^2 + y^2 + 1)^2}\n",
|
||
|
|
"$$\n",
|
||
|
|
"$$\n",
|
||
|
|
"\\frac{\\partial f}{\\partial y} {(x, y)} = \\frac{(x^2 + y^2 + 1) - (x^2 + y)(2y)}{(x^2 + y^2 + 1)^2} = \\frac{x^2 + y^2 + 1 - 2y(x^2 + y)}{(x^2 + y^2 + 1)^2}\n",
|
||
|
|
"$$"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"kernelspec": {
|
||
|
|
"display_name": "Python 3 (ipykernel)",
|
||
|
|
"language": "python",
|
||
|
|
"name": "python3"
|
||
|
|
},
|
||
|
|
"language_info": {
|
||
|
|
"codemirror_mode": {
|
||
|
|
"name": "ipython",
|
||
|
|
"version": 3
|
||
|
|
},
|
||
|
|
"file_extension": ".py",
|
||
|
|
"mimetype": "text/x-python",
|
||
|
|
"name": "python",
|
||
|
|
"nbconvert_exporter": "python",
|
||
|
|
"pygments_lexer": "ipython3",
|
||
|
|
"version": "3.9.2"
|
||
|
|
},
|
||
|
|
"toc": {
|
||
|
|
"base_numbering": 1,
|
||
|
|
"nav_menu": {},
|
||
|
|
"number_sections": false,
|
||
|
|
"sideBar": true,
|
||
|
|
"skip_h1_title": false,
|
||
|
|
"title_cell": "Table of Contents",
|
||
|
|
"title_sidebar": "Contents",
|
||
|
|
"toc_cell": false,
|
||
|
|
"toc_position": {},
|
||
|
|
"toc_section_display": true,
|
||
|
|
"toc_window_display": true
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 5
|
||
|
|
}
|